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  1. ❖✉t❣r♦✇t❤s ♦❢ t❤❡ ❉▲▼❋ Pr♦❥❡❝t✿ P❛rt ✷ ✿ ◆■❙❚ ❉✐❣✐t❛❧ ❘❡♣♦s✐t♦r② ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛❡ ❍♦✇❛r❞ ❈♦❤❧✯✱ ▼❛r❥❡ ▼❝❈❧❛✐♥✯✱ ❇♦♥✐t❛ ❙❛✉♥❞❡rs✯✱ ▼♦r✐t③ ❙❝❤✉❜♦t③ § ✯❆♣♣❧✐❡❞ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❛t❤❡♠❛t✐❝s ❉✐✈✐s✐♦♥✱ ◆■❙❚✱ ●❛✐t❤❡rs❜✉r❣✱ ▼❛r②❧❛♥❞✱ ❯✳❙✳❆✳ § ❉❛t❛❜❛s❡ ❙②st❡♠s ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ▼❛♥❛❣❡♠❡♥t ●r♦✉♣✱ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❇❡r❧✐♥ ❈❤❛❧❧❡♥❣❡s ✐♥ ✷✶st ❈❡♥t✉r②✿ ❊①♣❡r✐♠❡♥t❛❧ ▼❛t❤❡♠❛t✐❝❛❧ ❈♦♠♣✉t❛t✐♦♥ ■♥st✐t✉t❡ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❊①♣❡r✐♠❡♥t❛❧ ❘❡s❡❛r❝❤ ✐♥ ▼❛t❤❡♠❛t✐❝s✱ Pr♦✈✐❞❡♥❝❡✱ ❘❤♦❞❡ ■s❧❛♥❞ ❏✉❧② ✷✶✱ ✷✵✶✹ ✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✶ ✴ ✶✸

  2. ❉✐❣✐t❛❧ ❘❡♣♦s✐t♦r② ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛❡ ❖♥❧✐♥❡ ❝♦♠♣❡♥❞✐✉♠ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛❡ ♦rt❤♦❣♦♥❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥ ❢♦r♠✉❧❛❡ ❉❘▼❋ ❛tt❡♠♣ts t♦ ✉s❡ ❲❡❜ ✷✳✵ t❡❝❤♥♦❧♦❣✐❡s t♦ ♠♦✈❡ ❜❡②♦♥❞ t❤❡ st❛t✐❝ ♣r❡s❡♥t❛t✐♦♥ ♦❢ r❡❢❡r❡♥❝❡ ❞❛t❛ t♦ ❛ ♣❧❛t❢♦r♠ t❤❛t ❡♥❝♦✉r❛❣❡s ❝♦♠♠✉♥✐t② ✐♥t❡r❛❝t✐♦♥ ❛♥❞ ❝♦❧❧❛❜♦r❛t✐♦♥ ✳ ❉❘▼❋ ✉t✐❧✐③❡s ♦❢ ❉▲▼❋ ▲ ❆ ❚ ❊❳ ♠❛❝r♦s t✐❡ s♣❡❝✐✜❝ ❝❤❛r❛❝t❡r s❡q✉❡♥❝❡s t♦ ✇❡❧❧✲❞❡✜♥❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝ts ✳ Pr♦✈✐❞❡s ❛♥ ✐♥t❡r♥❡t ❧✐♥❦ t♦ st❛♥❞❛r❞ ✱ ♣r❡❝✐s❡ ♦rt❤♦❣♦♥❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥ ❞❡✜♥✐t✐♦♥s t❤r♦✉❣❤ t❤❡ ❉▲▼❋ ❛♥❞ ❉❘▼❋ ❯s❡s ▼❡❞✐❛❲✐❦✐ ✇✐❦✐ s♦❢t✇❛r❡ ▼❛t❤▼▲ s✉♣♣♦rt ▲ ❚ ❊❳▼▲ ❆ ▼❛t❤❏❛① ✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✷ ✴ ✶✸

  3. ❉❘▼❋ ❣♦❛❧s ❚❤❡ ❉❘▼❋ ✇✐❧❧ ❜❡ ❞❡s✐❣♥❡❞ ❢♦r ❛ ♠❛t❤❡♠❛t✐❝❛❧❧② ❧✐t❡r❛t❡ ❛✉❞✐❡♥❝❡ ❛♥❞ s❤♦✉❧❞✿ ✶ ❢❛❝✐❧✐t❛t❡ ✐♥t❡r❛❝t✐♦♥ ❛♠♦♥❣ ❛ ❝♦♠♠✉♥✐t② ♦❢ ♠❛t❤❡♠❛t✐❝✐❛♥s ❛♥❞ s❝✐❡♥t✐sts ✐♥t❡r❡st❡❞ ✐♥ ❢♦r♠✉❧❛❡ ❞❛t❛ r❡❧❛t❡❞ t♦ ♦rt❤♦❣♦♥❛❧ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❖P❙❋✮❀ ✷ ❜❡ ❡①♣❛♥❞❛❜❧❡✱ ❛❧❧♦✇✐♥❣ t❤❡ ✐♥♣✉t ♦❢ ♥❡✇ ❢♦r♠✉❧❛❡❀ ✸ ❜❡ ❛❝❝❡ss✐❜❧❡ ❛s ❛ st❛♥❞❛❧♦♥❡ r❡s♦✉r❝❡ ❀ ✹ ❤❛✈❡ ❛ ✉s❡r ❢r✐❡♥❞❧②✱ ❝♦♥s✐st❡♥t✱ ❛♥❞ ❤②♣❡r❧✐♥❦❛❜❧❡ ✈✐❡✇♣♦✐♥t ❛♥❞ ❛✉t❤♦r✐♥❣ ♣❡rs♣❡❝t✐✈❡ ❀ ❛♥❞ ✺ ❝♦♥t❛✐♥ ❡❛s✐❧② s❡❛r❝❤❛❜❧❡ ♠❛t❤❡♠❛t✐❝s ❛♥❞ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ♠♦❞❡r♥ ▼❛t❤▼▲ t♦♦❧s ❢♦r ❡❛s② t♦ r❡❛❞ ✱ s❝❛❧❛❜❧② r❡♥❞❡r❡❞ ♠❛t❤❡♠❛t✐❝s ✳ ✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✸ ✴ ✶✸

  4. ❉❘▼❋ ❙❡❡❞✐♥❣ Pr♦❥❡❝ts ▼❛t❤ ❖❈❘✱ ▼❛❝r♦ r❡♣❧❛❝❡♠❡♥t✱ ❛♥❞ ❲✐❦✐t❡①t ❣❡♥❡r❛t✐♦♥✱ t♦ ✐♠♣❧❡♠❡♥t ♣r❡✲❡①✐st✐♥❣ ❜♦♦❦ ❝♦♠♣❡♥❞✐❛ ▼❛t❤❡♠❛t✐❝❛❧ ❖♣t✐❝❛❧ ❈❤❛r❛❝t❡r ❘❡❝♦❣♥✐t✐♦♥ ♣r♦❥❡❝t ❇❛t❡♠❛♥ ♠❛♥✉s❝r✐♣t ♣r♦❥❡❝t ✿ ❍✐❣❤❡r ❚r❛♥s❝❡♥❞❡♥t❛❧ ❋✉♥❝t✐♦♥s✱ ❚❛❜❧❡s ♦❢ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s ❇②r❞ ✫ ❋r✐❡❞♠❛♥ ✬s ❍❛♥❞❜♦♦❦ ♦❢ ❊❧❧✐♣t✐❝ ■♥t❡❣r❛❧s ❢♦r ❊♥❣✐♥❡❡rs ❛♥❞ ❙❝✐❡♥t✐sts ❉▲▼❋ ▲ ❆ ❚ ❊❳ ♠❛❝r♦ r❡♣❧❛❝❡♠❡♥t ♣r♦❥❡❝t ❍②♣❡r❣❡♦♠❡tr✐❝ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ❛♥❞ ❚❤❡✐r q ✲❆♥❛❧♦❣✉❡s ✕ ❑▲❙ ❑▲❙ ❛❞❞❡♥❞✉♠ ❜② ❚♦♠ ❑♦♦r♥✇✐♥❞❡r ❢✉t✉r❡❄✿ ❆♥❞r❡✇s✱ ❆s❦❡② ✫ ❘♦② ✿ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s ❢✉t✉r❡❄✿ ■s♠❛✐❧ ✿ ❈❧❛ss✐❝❛❧ ❛♥❞ ◗✉❛♥t✉♠ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ✐♥ ❖♥❡ ❱❛r✐❛❜❧❡ ❢✉t✉r❡❄✿ ❡t❝✳ ❲✐❦✐t❡①t ❣❡♥❡r❛t✐♦♥ ♣r♦❥❡❝t ◆■❙❚ ❉✐❣✐t❛❧ ▲✐❜r❛r② ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❋✉♥❝t✐♦♥s ✭❝❤✳ ✷✺✮ ✿ ✶✼✵ ❢♦r♠✉❧❛s ✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✹ ✴ ✶✸

  5. ❉❘▼❋ ❩❡t❛ ❛♥❞ ❘❡❧❛t❡❞ ❋✉♥❝t✐♦♥s P❛❣❡ ✳✳✳ ❉▲▼❋ ❲✐❦✐t❡①t · · · · · · · · · · · · · · · · · · ✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✺ ✴ ✶✸

  6. ❉▲▼❋ ♠❛❝r♦s ♣r♦✈✐❞❡ s❡♠❛♥t✐❝ ❝♦♥t❡♥t ✐♥ ❢♦r♠✉❧❛s ❉▲▼❋ ❖P❙❋ ▼❛❝r♦s ✈✐❛ ▲ ❆ ❚ ❊❳▼▲✲s❡r✈❡r ✺✹✻ s❡♠❛♥t✐❝ ❉▲▼❋ ▲ ❆ ❚ ❊❳ ❖P❙❋ ♠❛❝r♦s ❛❞❞✐t✐♦♥❛❧ ✸✽ s❡♠❛♥t✐❝ ▲ ❚ ❊❳ ♠❛❝r♦s ❆ ❖❜❥❡❝ts✿ ❭s✉♠✱❭✐♥t✱❭❞❡r✐✈④❢⑥④①⑥✱❭q❞❡r✐✈❬♥❪④q⑥❅④③⑥ ❈♦♥st❛♥ts✿ ❭❡①♣❡✱❭✐✉♥✐t✱❭❝♣✐✱❭❊✉❧❡r❈♦♥st❛♥t ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s ❛♥❞ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s Γ( z ) ❭❊✉❧❡r●❛♠♠❛❅④③⑥ ❤tt♣✿✴✴❞❧♠❢✳♥✐st✳❣♦✈✴✺✳✸✵★❊✶ J ν ( z ) ❭❇❡ss❡❧❏④❭♥✉⑥❅④③⑥ ❤tt♣✿✴✴❞❧♠❢✳♥✐st✳❣♦✈✴✶✵✳✷★❊✷ Q µ ν ( z ) ❭▲❡❣❡♥❞r❡◗❬❭♠✉❪④❭♥✉⑥❅④③⑥ ✿ ❤tt♣✿✴✴❞❧♠❢✳♥✐st✳❣♦✈✴✶✹✳✸★❊✼ P ( α,β ) ( x ) ❭❏❛❝♦❜✐P④❭❛❧♣❤❛⑥④❭❜❡t❛⑥④♥⑥❅④①⑥ ❤tt♣✿✴✴❞❧♠❢✳♥✐st✳❣♦✈✴✶✽✳✸★❚✶✳t✶✳r✸ n ✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✻ ✴ ✶✸

  7. ▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛❡ ❲❤❡r❡❛s ❲✐❦✐♣❡❞✐❛ ❛♥❞ ♦t❤❡r ✇❡❜ ❛✉t❤♦r✐♥❣ t♦♦❧s ♠❛♥✐❢❡st ♥♦t✐♦♥s ♦r ❞❡s❝r✐♣t✐♦♥s ❛s ✜rst ❝❧❛ss ♦❜❥❡❝ts ✱ t❤❡ ❉❘▼❋ ❞♦❡s t❤❛t ✇✐t❤ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛❡ ✳ ❉❘▼❋ ♣r♦✈✐❞❡s ❢♦r ❡❛❝❤ ❢♦r♠✉❧❛✱ ❛ ❢♦r♠✉❧❛ ❤♦♠❡ ♣❛❣❡ ✿ ✶ ❘❡♥❞❡r❡❞ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢♦r♠✉❧❛ ✭r❡q✉✐r❡❞✮❀ ✷ ❈♦♥str❛✐♥ts t❤❡ ❢♦r♠✉❧❛ ♠✉st ♦❜❡② ✸ ❙✉❜st✐t✉t✐♦♥s r❡q✉✐r❡❞ t♦ ✉♥❞❡rst❛♥❞ ❢♦r♠✉❧❛❀ ✹ ❇✐❜❧✐♦❣r❛♣❤✐❝ ❝✐t❛t✐♦♥ ✭r❡q✉✐r❡❞✮❀ ✺ ❖♣❡♥ s❡❝t✐♦♥ ❢♦r ♣r♦♦❢s ✭r❡q✉✐r❡❞✮ ✕ ❉▲▼❋ ❀ ✻ ▲✐st ♦❢ s②♠❜♦❧s ❛♥❞ ❧✐♥❦s t♦ ❞❡✜♥✐t✐♦♥s ✭r❡q✉✐r❡❞✮ ✕ ❉▲▼❋ ♠❛❝r♦s ❀ ✼ ❖♣❡♥ s❡❝t✐♦♥ ❢♦r ♥♦t❡s ✕ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❢♦r♠✉❧❛s ❀ ❛♥❞ ✽ ❖♣❡♥ s❡❝t✐♦♥ ❢♦r ❡①t❡r♥❛❧ ❧✐♥❦s ✕ ❝♦♠♣✉t❡r ❣❡♥❡r❛t❡❞ ♣r♦♦❢s ❀ ✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✼ ✴ ✶✸

  8. ❙❛♠♣❧❡ ❢♦r♠✉❧❛ ❤♦♠❡ ♣❛❣❡ ✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✽ ✴ ✶✸

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